Can I keep focus in a showdown and play a reaction immediately after playing my action spell?

Yes — after you play an Action spell, you keep priority and can immediately add a Reaction (or several) on top before passing.

Adding a chain item gives you priority back, so you can chain reactions behind your own action. Two things to keep in mind: you're holding priority, not focus — once the chain you opened with focus closes, focus passes to the next player in turn order (Rule 347.1.b). And your opponent still gets a full response window once you pass, so this doesn't stop them from answering your original spell.